Introduction to Graphs Minors, Tree Decompositions and FPT Algorithms. MPRI February 2, 2021 Contents 1 Introduction 3 1.1 Basic Definitions and Terminology . .3 1.2 Three Algorithmic Problems . .3 1.3 Minors . .5 2 Minor Closed Classes and Well Quasi Orders 9 2.1 Wagner’s Conjecture - Well Quasi Orders . .9 2.2 Words, Paths and trees . 10 3 Tree Width 15 3.1 Definitions . 15 3.2 Treewidth and Minors . 16 3.3 Digression : Hadwiger Conjecture . 18 3.4 Separators . 18 3.5 Duality - Cops and Robbers . 20 3.6 Treewidth and Planar Graphs . 22 3.7 Digression : Erdos-Posa˝ Property . 23 3.8 Wagner’s Conjecture . 24 3.9 Rooted disjoint path problem and minor detection . 25 3.10 Tree Decompositions of Planar Graphs . 26 4 Tree Width and Algorithms 28 4.1 Algorithms on Trees . 28 4.2 Algorithms for bounded-treewidth graphs . 29 4.3 Monadic Second Order Logic . 31 4.4 Computing tree width . 33 4.5 FPT Algorithms . 34 4.5.1 Existence of FPT via Robertson and Seymour Theorems . 35 4.5.2 TreeWidth Based Algorithms . 35 4.5.3 Branching . 36 4.5.4 Kernelization . 36 4.5.5 Iterative Compression . 37 4.5.6 Colour Coding . 37 1 Introduction These are notes in complement to a 12h lecture given in 2020-2021 in the MPRI (Master Parisien de Recherche en Informatique) about minors, treewidth and FPT algorithms on bounded treewidth classes of graphs. For a general textbook on graph theory, the book of Bondy and Murty ([4] is an excellent reading, but the theory of graphs minors and tree decomposition is not deeply discussed. On this particular topic, there exists a good a short introduction by Lovasz ([11]), but the main influence for this course is Chapter 12 of the book of Diestel ([7]), which is also a good source for exercises. For Fixed Parameter Tractable Algorithms, a classic book is ([8]). 2 Chapter 1 Introduction 1.1 Basic Definitions and Terminology In this course, a graph is given by a set V, whose elements are called vertices, and a set E whose elements, called edges of the graphs, are distinct subsets of size 2 of V. According to the usual vocabulary, this means that our graph will always be simple and without loops. Unless specified, V will always be a finite set. For a graph G, V(G) will always denote its set of vertices, E(G) its set of edges. Very often we will write xy instead of fx; yg for an edge of G. A vertex v is a neighbour of a vertex u if uv 2 E(G). The neighbourhood of u, denoted N(u) is the set of neighbours of u. Its degree, denoted d(u) is the cardinality of its neighbourhood. The maximum degree of a graph is usually denoted ∆. A graph with no edges will be called a stable set, or independent set, and a graph will all possible edges between its vertices a clique, or complete graph. The complete graph on n vertices is usually denoted Kn. The path Pk is a graph with V(Pk) = fx1; x2;:::; xkg, with edges E = fxi xi+1; 1 ≤ i ≤ k − 1g. The vertices x1 and xk are called the endpoints of the path. If we add the edge xk x1 to Pk then the resulting graph is the circuit on k vertices, denoted Ck. 1.2 Three Algorithmic Problems Consider the following problem of connectivity. Problem : k disjoint path problem Input : A graph G, an integer k and two subsets of vertices A and B of size k Output : TRUE if there exists k vertex disjoint paths from A to B? This problem is a very classical one, and Ford-Fulkerson Algorithm tells us that this is solvable in time O((kjE(G)j) (classical Ford-Fulkerson Algorithm is for edge disjoint path in the directed case, but it is easy to reduce our case to this one). The maximum value k corresponds to a minimum vertex cut separating A and B and is a classical result of Menger. Theorem 1.1 (Menger,1927,[12]) Let x and y be distinct vertices of a graph G. Then the minimum number of vertices whose deletion separates x from y is equal to the maximum number of internally disjoint paths between 3 x and y. Proof. See [4]. Now consider the smilingly similar problem. Problem : k-disjoint rooted path problem Input : A graph G, an integer k, and two subsets of vertices X = fx1; x2;:::; xkg and Y = fy1; y2;:::; ykg Output : TRUE if there exists disjoint paths P1; P2;:::; Pk, such that Pi is a path from xi to yi. This kind of problem in a more general form is known as commodity flow problem and has many applications. With k part of the input, this problem is NP-complete, even restricter to the class of planar graphs. Nevertheless, in the Graph Minor series of papers, Robertson and Seymour proved a polynomial algorithm for fixed k. This result is extremely difficult and relies and techniques and notions that will be illustrated in this course. Theorem 1.2 (Robertson-Seymour, [13]) The k-disjoint path problem can be solved in time O ( f (k):n3 The result has been improved to quadratic time by Kawabayarashi, Kobashi and Reed ([]). Let us see an algorithmic consequence of this result related to topological minor detection. Definition 1.3 A graph H is topological minor of a graph G if there exists a injective mapping f from V(H) to V(G) such that for each edge uv of H, there exists in G a path Puv connecting f (u) and f (v) in G with the property that all these path are internally disjoint. Example. Describe the graphs that do not contain the following graphs as topological minors : K3, K1;3, K1;4. A natural algorithmic problem is then the following. Problem : Topological H-minor detection Input : A graph G and a graph H. Output : TRUE if H is a topological minor of G, FALSE otherwise. The problem is NP-complete if H is part of the input, but if H but if H is fixed, then this problem was proven to be polynomial by Robertson and Seymour. Theorem 1.4 Let H be a fixed graph. There exists a polynomial time algorithm to decide whether H is a topological minor of a given graph G. Proof. Let f : V(H) ! V(G) be an injection (note there are polynomially many such objects), we want to decide if there exists disjoint paths in G between the f (v) corresponding to edges of H. To do that, we replace each vertex f (v) by dH(v) copies of f (v) (having the 4 same neighbours). Now, for k = jE(H)j, solving the k-Rooted Disjoint Path Problem for these sources clearly solves the desired question. The complexity of this algorithm is hence O( f (k)) nk, where k is the size of H, and n the size of G. It is therefore polynomial for every fixed k. In 2010, Grohe, Kawabarayashi, Marx, and Wollan proved a stronger result, that this can be done in O( f (k)) n3. Such an algorithm is called Fixed Parameter Tractable (FPT) algorithm. We will discuss more about those in the last chapter of this course. In particular, the previous theorem implies that any family of graphs that is defined with forbidding a FINITE family of graphs as topological minors is polynomially testable. One such family is very well known, it is the family of planar graphs, as was proven by Kuratowski in 1930. Theorem 1.5 (Kuratowski,1930) A graph G is planar if and only if it does not contain K5 or K3;3 as a topological minor. Planar graph will play a central role in this course, and as we will see, this theorem will be characteristic of the kind of result we will be interested. The crucial fact here is not that planar graphs are defined by a certain list of forbidden topological minors, this is easy (why?), it is the finiteness of this list that is non trivial. The central result of Robertson and Seymour theory is that many different graph properties can be characterised by a finite list of forbidden substructures, and hence get polynomial time recognition algorithm. Of course, one does not need the difficult of Robertson and Seymour to prove that planar graphs are polynomially recognisable, there even exists linear time algorithm to do that. Nevertheless, we will see later that there are instances of such recognition problem for which the only proof of polyniomiality was obtained through their results. 1.3 Minors We define three operations on a graph G (at the end of each line the notation for the resulting graph). 1. Remove a vertex v (and all its incident edges) : G n v 2. Remove an edge e (but not its end vertices) : G n e 3. Contract an edge e = xy, which means remove x and y, add a new vertex z whose neigh- bourhood is the union of the neighbourhoods of x and y (without putting any loop on z): G=e. A contraction G=e is topological if one of the endpoints of e has degree 2. Its inverse is the subdivision operation which consists in removing an edge xy, adding a new vertex z, and adding the edges xz and zy. Definition 1.6 Let G and H be two graphs. • H is an induced subgraph of G if H is obtained from G by the repeated use of rule 1.